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Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations

Ball, J. M. (1997) Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations. Journal of Nonlinear Science, 7 (5). pp. 475-502. ISSN 0938-8974 (Paper) 1432-1467 (Online)

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Abstract

A class of semiflows having possibly nonunique solutions is defined. The measurability and continuity properties of such generalized semiflows are studied. It is shown that a generalized semiflow has a global attractor if and only if it is pointwise dissipative and asymptotically compact. The structure of the global attractor in the presence of a Lyapunov function, and its connectedness and stability properties are studied. In particular, examples are given in which the global attractor is a single point but is not Lyapunov stable.

The existence of a global attractor for the 3D incompressible Navier-Stokes equations is established under the (unproved) hypothesis that all weak solutions are continuous from $(0,\infty)$ to $L^2$.

Item Type:Article
Additional Information:There is a one-page erratum to this paper. http://www.springerlink.com/openurl.asp?genre=article&eissn=1432-1467&volume=8&issue=2&spage=233
Subjects:O - Z > Partial differential equations
D - G > Dynamical systems and ergodic theory
D - G > Fluid mechanics
Research Groups:Oxford Centre for Nonlinear PDE
ID Code:197
Deposited By:John Ball
Deposited On:30 Aug 2005
Last Modified:20 Jul 2009 14:19

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